A Functional Inequality in Restricted Domains of Banach Modules

doi:10.1155/2009/973709

Research Article

A Functional Inequality in Restricted Domains of Banach Modules

M. B. Moghimi,

Abbas Najati,

and Choonkil Park

1Department of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili, Ardabil 56199–11367, Iran

2Department of Mathematics, Hanyang University, Seoul 133-791, South Korea

Correspondence should be addressed to Choonkil Park,[email protected] Received 28 April 2009; Revised 2 August 2009; Accepted 16 August 2009 Recommended by Binggen Zhang

We investigate the stability problem for the following functional inequalityαfxy/2α βfyz/2β γfzx/2γ ≤ fxyzon restricted domains of Banach modules over aC^∗-algebra. As an application we study the asymptotic behavior of a generalized additive mapping.

Copyrightq2009 M. B. Moghimi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction and Preliminaries

The following question concerning the stability of group homomorphisms was posed by Ulam1: Under what conditions does there exist a group homomorphism near an approximate group homomorphism?

Hyers2considered the case of approximately additive mappingsf :E → E, where EandEare Banach spaces andfsatisfies Hyers inequality

f xy

−fx−f

y≤ε 1.1

for allx, y∈E.

In 1950, Aoki 3 provided a generalization of the Hyers’ theorem for additive mappings and in 1978, Rassias4generalized the Hyers’ theorem for linear mappings by allowing the Cauchy difference to be unboundedsee also5. The result of Rassias’ theorem has been generalized by Forti6,7and Gavruta8who permitted the Cauchy difference to be bounded by a general control function. During the last three decades a number of papers

have been published on the generalized Hyers-Ulam stability to a number of functional equations and mappingssee9–23. We also refer the readers to the books24–28.

Throughout this paper, letAbe a unitalC^∗-algebra with unitary groupUA, unite, and norm| · |. Assume thatXis a leftA-module andYis a left BanachA-module. An additive mappingT :X → Yis calledA-linear ifTax aTxfor alla∈ Aand allx ∈X. In this paper, we investigate the stability problem for the following functional inequality:

αf xy

2α

βf yz

2β

γf zx

2γ

≤f

xyz 1.2

on restricted domains of Banach modules over aC^∗-algebra, whereα, β, γare nonzero positive real numbers. As an application we study the asymptotic behavior of a generalized additive mapping.

2. Solutions of the Functional Inequality 1.2

Theorem 2.1. LetXandMbe leftA-modules and letα, β, γ be nonzero real numbers. If a mapping f:X → Mwithf0 0 satisfies the functional inequality

αf

axay 2α

βf

ayaz 2β

γaf

zx 2γ

≤f

axayaz 2.1

for allx, y, z∈Xand alla∈UA, thenfisA-linear.

Proof. Lettingz−x−yin2.1,we get

αf

axay 2α

βf

−ax 2β

γaf

−y 2γ

0 2.2

for allx, y∈Xand alla∈UA. Lettingx0resp.,y0in2.2, we get

αfay 2α

γaf

−y 2γ

0,

resp., αfax 2α

βf

−ax 2β

0

2.3

for allx, y ∈ Xand alla ∈ UA. Hencefay −γ/αaf−α/γyand it follows from 2.2and2.3that andfaxay/2α−fax/2α−fay/2α 0 for allx, y ∈Xand all a∈UA.Thereforefxy fx fyfor allx, y∈X.Hencefrx rfxfor allx∈X and all rational numbersr.

Now leta∈Aa /0and letmbe an integer number withm >4|a|. Then by Theorem 1 of29, there exist elementsu1, u2, u3 ∈ UAsuch that3/ma u1u2u3. Sincef is

additive and frbx −γ/αrbf−α/γx for all x ∈ X,all rational numbers r and all b∈UA, we have

fax m 3f

3 max

m

3fu1xu2xu3x m

3 fu1x fu2x fu3x −m

3 γ

αu1u2u3f

−α γx

−m

3 γ α

3 maf

−α γx

−γ

αaf

−α γx

2.4

for allx∈X. Replacing−γ/αxinstead ofxin the above equation, we have

f

−γ αax

−γ

αafx 2.5

for allx ∈ X.Sinceais an arbitrary nonzero element inAin the previous paragraph, one can replace−α/γainstead ofain2.5. Thus we havefax afxfor allx ∈Xand all a∈Aa /0.Sof:X → YisA-linear.

The following theorem is another version ofTheorem 2.1on a restricted domain when α, β, γ >0.

Theorem 2.2. LetXand Mbe leftA-modules and letd, α, β, γ be nonzero positive real numbers.

Assume that a mappingf : X → Msatisfiesf0 0 and the functional inequality2.1for all x, y, z∈Xwithxyz ≥dand alla∈UA. ThenfisA-linear.

Proof. Lettingz−x−ywithxy ≥din2.1, we get

αf

axay 2α

βf

−ax 2β

γaf

−y 2γ

0 2.6

for alla∈UA. Letδmax{|β|⁻¹d,|γ|⁻¹d}and letxy ≥δ.Then

βxγy≥minβ,γxy≥minβ,γδ≥d. 2.7 Therefore replacingxandyby 2βxand 2γyin2.6, respectively, we get

αf

βaxγay α

βf−ax γaf

−y

0 2.8

for allx, y∈Xwithxy ≥δand alla∈UA.

Similar to the proof of Theorem 3 of30 see also31, we prove thatfsatisfies2.8 for allx, y ∈ Xand alla ∈ UA. Supposexy < δ.Ifxy 0, letz ∈ Xwith zδ,otherwise

z:

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

δx x

x, ifx ≥y; δy y

y, ify≥ x.

2.9

Sinceα, β, γ >0,it is easy to verify that

2β⁻¹γ

zβ⁻¹γyβγ⁻¹x−

12βγ⁻¹ z≥δ, xz ≥δ,

2

1β⁻¹γ

zy≥δ, 2

1β⁻¹γ

zβγ⁻¹x−

12βγ⁻¹ z≥δ,

2β⁻¹γ

zβ⁻¹γyz ≥δ.

2.10

Therefore

αf

βaxγay α

βf−ax γaf

−y

αf

βaxγay α

βf

−

2β⁻¹γ

az−β⁻¹γay

γaf

12βγ⁻¹

z−βγ⁻¹x

αf

βaxγaz α

βf−ax γaf−z

αf 2

βγ

azγay α

βf

−2

1β⁻¹γ az

γaf

−y

−

αf

βaxγaz α

βf

−2

1β⁻¹γ az

γaf

12βγ⁻¹

z−βγ⁻¹x

−

αf 2

βγ

azγay α

βf

−

2β⁻¹γ

az−β⁻¹γay

γaf−z

0.

2.11 Hencefsatisfies2.8and we infer thatfsatisfies2.2for allx, y∈Xand alla∈UA. By Theorem 2.1,fisA-linear.

3. Generalized Hyers-Ulam Stability of 1.2 on a Restricted Domain

In this section, we investigate the stability problem forA-linear mappings associated to the functional inequality 1.2 on a restricted domain. For convenience, we use the following abbreviation for a given functionf:X → Yanda∈UA:

Daf x, y, z

:αf

axay 2α

βf

ayaz 2β

γaf

zx 2γ

3.1

for allx, y, z∈X.

Theorem 3.1. Letd, α, β, γ >0, p∈0,1,andθ, ε≥0 be given. Assume that a mappingf :X → Ysatisfies the functional inequality

fDaf

x, y, z≤f

axayazθε

x^py^pz^p

3.2

for allx, y, z∈Xwithxyz ≥ dand alla∈UA. Then there exist a uniqueA-linear mappingT :X → Yand a constantC >0 such that

fx−Tx≤C24×2^pα^p−1ε

2−2^p x^p 3.3

for allx∈X.

Proof. Letz−x−ywithxy ≥d. Then3.2implies that αf

axay 2α

βf

−ax 2β

γaf

−y 2γ

≤f0θε

x^py^pxy^p

≤f0θ2ε

x^py^p .

3.4

Thus αf

axay α

βf

−ax β

γaf

−y γ

≤f0θ2^p1ε

x^py^p

3.5

for all x, y ∈ X with xy ≥ d and all a ∈ UA. Let δ max{β⁻¹d, γ⁻¹d} and let xy ≥δ.Thenβxγy ≥d.Therefore it follows from3.5that

αf

βaxγay α

βf−ax γaf

−y≤f0θ2^p1εβx^pγy^p

3.6

for allx, y ∈ Xwith xy ≥ δ and all a ∈ UA. For the casexy < δ, letzbe an element ofXwhich is defined in the proof ofTheorem 2.2. It is clear thatz ≤2δ.Using 2.11and3.6, we get

αf

βaxγay α

βf−ax γaf

−y

≤

αf

βaxγay α

βf

−

2β⁻¹γ

az−β⁻¹γay

γaf

12βγ⁻¹

z−βγ⁻¹x

αf

βaxγaz α

βf−ax γaf−z

αf 2

βγ

azγay α

βf

−2

1β⁻¹γ az

γaf

−y

αf

βaxγaz α

βf

−2

1β⁻¹γ az

γaf

12βγ⁻¹

z−βγ⁻¹x

αf 2

βγ

azγay α

βf

−

2β⁻¹γ

az−β⁻¹γay

γaf−z

≤5f0θ

4^p1εδ^p 2

2βγ_p 2^p

βγ_p γ^p

6×2^pεβx^pγy^p 3.7

for allx, y∈Xwithxy< δand alla∈UA. Hence αf

βaxγay α

βf−ax γaf

−y≤K6×2^pεβx^pγy^p

3.8

for allx, y∈Xand alla∈UA, where

K:5f0θ

4^p1εδ^p 2

2βγp2^p

βγpγ^p

. 3.9

Lettingx0 andy0 in3.8, respectively, we get αfγay

α

βf0 γaf

−y≤K6×2^pεγy^p, αf

βax α

βf−ax γaf0

≤K6×2^pεβx^p 3.10

for allx, y∈Xand alla∈UA. It follows from3.8and3.10that f

xy

−fx−f

y≤α⁻¹ βγf03K12×2^pε

αx^pαy^p

3.11

for all x, y ∈ X. By the results of Hyers2 and Rassias4, there exists a unique additive mappingT :X → Ygiven byTx lim_{n→ ∞}2⁻ⁿf2ⁿxsuch that

fx−Tx≤α⁻¹ βγf03K

24×2^pα^p−1ε

2−2^p x^p 3.12

for allx∈X. It follows from the definition ofT and3.2thatT0 0 andDaTx, y, z ≤ Taxayazfor allx, y, z∈Xwithxyz ≥dand alla∈UA. HenceT is A-linear byTheorem 2.2.

We apply the result ofTheorem 3.1to study the asymptotic behavior of a generalized additive mapping. An asymptotic property of additive mappings has been proved by Skof 32 see also30,33.

Corollary 3.2. Letα, β, γbe nonzero positive real numbers. Assume that a mappingf:X → Ywith f0 0 satisfies

Daf x, y, z

−f

axayaz−→0 asxyz −→ ∞ 3.13 for alla∈UA,thenfisA-linear.

Proof. It follows from3.13that there exists a sequence {δn},monotonically decreasing to zero, such that

Daf x, y, z

−f

axayaz≤δn 3.14

for allx, y, z∈Xwithxyz ≥nand alla∈UA. Therefore Daf

x, y, z≤f

axayazδn 3.15

for allx, y, z∈Xwithxyz ≥nand alla∈UA. Applying3.15andTheorem 3.1, we obtain a sequence{Tn:X → Y}of uniqueA-linear mappings satisfying

fx−Tnx≤15α⁻¹δn 3.16

for allx∈X. Since the sequence{δn}is monotonically decreasing, we conclude

fx−Tmx≤15α⁻¹δm≤15α⁻¹δn 3.17 for allx∈Xand allm≥n.The uniqueness ofTnimpliesTmTnfor allm≥n.Hence letting n → ∞in3.16, we obtain thatfisA-linear.

The following theorem is another version ofTheorem 3.1for the casep >1.

Theorem 3.3. Letp >1, d >0, ε≥0 be given and letα, β, γbe nonzero real numbers. Assume that a mappingf :X → Ywithf0 0 satisfies the functional inequality

Daf

x, y, z≤f

axayazε

x^py^pz^p

3.18 for allx, y, z∈Xwithxyz ≤dand alla∈UA. Then there exists a uniqueA-linear mappingφ:X → Ysuch that

φx−fx≤ 62^p×2^p|α|^p−1ε

2^p−2 x^p 3.19

for allx∈Xwithx ≤d/8|α|andφx lim_{n→ ∞}2ⁿf2⁻ⁿx.

Proof. Lettingz−x−yin3.18, we get αf

axay 2α

βf

−ax 2β

γaf

−y 2γ

≤ε

x^py^pxy^p

3.20

for allx, y∈Xwithxy ≤d/2 and alla∈UA. Hence αf

axay α

βf

−ax β

γaf

−y γ

≤2^pε

x^py^pxy^p

3.21

for allx, y∈Xwithxy ≤d/4 and alla∈UA. It follows from3.21that αfax

α βf

−ax β

≤2^p1εx^p, αfay

α

γaf

−y γ

≤2^p1εy^p 3.22

for allx, y∈Xwithx,y ≤d/4 and alla∈UA. Adding3.21to3.22, we get αf

axay α

−αfax α

−αfay α

≤2^pε

3x^p3y^pxy^p

3.23

for allx, y∈Xwithx,y ≤d/8 and alla∈UA. Therefore f

xy

−fx−f

y≤2^p|α|^p−1ε

3x^p3y^pxy^p

3.24 for allx, y ∈ Xwith x,y ≤ d/8|α|. Letx ∈ Xwithx ≤ d/8|α|. We may puty xin 3.24to obtain

f2x−2fx≤62^p×2^p|α|^p−1εx^p. 3.25

We can replace x byx/2ⁿ¹ in 3.25for all nonnegative integers n. Then using a similar argument given in4, we have

2ⁿ¹f 2⁻ⁿ⁻¹x

−2ⁿf2⁻ⁿx≤62^p× 2

2^p n

|α|^p−1εx^p. 3.26

Hence we have the following inequality:

2ⁿ¹f 2⁻ⁿ⁻¹x

−2^mf

2^−mx≤ ⁿ

km

2^k1f 2^−k−1x

−2^kf 2^−kx

≤62^p|α|^p−1ε n km

2 2^p

k

x^p

3.27

for all x ∈ X with x ≤ d/8|α|and all integers n ≥ m ≥ 0.SinceY is complete, 3.27 shows that the limitTx limn→ ∞2ⁿf2⁻ⁿxexists for allx∈Xwithx ≤ d/8|α|. Letting m 0 andn → ∞in3.27, we obtain thatT satisfies inequality3.19for allx ∈ Xwith x ≤d/8|α|. It follows from the definition ofT and3.24that

T xy

Tx T y

3.28

for allx, y∈Xwithx,y,xy ≤d/8|α|. Hence

Tx 2

1

2Tx 3.29

for allx∈Xwithx ≤d/8|α|. We extend the additivity ofTto the whole spaceXby using an extension method of Skof34. Letδ:d/8|α|andx∈Xbe given withx> δ.Letkkx be the smallest integer such that 2^k−1δ <x ≤2^kδ.We define the mappingφ:X → Yby

φx:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

Tx, if x ≤δ,

2^kT 2^−kx

, if x> δ.

3.30

Letx ∈Xbe given withx > δ and letk kxbe the smallest integer such that 2^k−1δ <

x ≤2^kδ.Thenk−1 is the smallest integer satisfying 2^k−2δ <x/2 ≤2^k−1δ.Ifk1, we have φx/2 Tx/2andφx 2Tx/2. Thereforeφx/2 1/2φx.For the casek > 1, it follows from the definition ofφthat

φx 2

2^k−1T

2^−k−1x 2

1 2 ·2^kT

2^−kx 1

2φx. 3.31

From the definition ofφand3.29, we get thatφx/2 1/2φxholds true for allx∈X.

Letx∈Xand letkbe an integer such thatx ≤2^kδ.Then φx 2^kφ

2^−kx 2^kT

2^−kx lim

n→ ∞2^nkf

2^−nkx lim

n→ ∞2ⁿf 2⁻ⁿx

. 3.32

It remains to prove thatφisA-linear. Letx, y ∈ Xand letnbe a positive integer such that x,y,xy ≤2ⁿδ.Sinceφx/2 1/2φxfor allx∈XandTsatisfies3.28, we have

φ xy

2ⁿφ xy

2ⁿ

2ⁿT xy

2ⁿ

2ⁿ Tx

2ⁿ

Ty 2ⁿ

2ⁿ

φx 2ⁿ

φy 2ⁿ

φx φ y

.

3.33

Henceφis additive. Sinceφx limn→ ∞2ⁿf2⁻ⁿxfor allx∈ X, we have from3.22that αφay/α γaφy/γfor all y ∈ Xand alla ∈ UA.Lettinga e, we get αφy/α γφy/γ. Thereforeφay aφy for ally ∈ Xand alla ∈ UA.This proves thatφ is A-linear. Also,φsatisfies inequality3.19for allx ∈Xwithx ≤d/8|α|, by the definition ofφ.

For the case p 1 we use the Gajda’s example 35 to give the following counterexample.

Example 3.4. Letφ:C → Cbe defined by

φx:

⎧⎨

⎩

x, for|x|<1,

1, for|x| ≥1. 3.34

Consider the functionf:C → Cby the formula

fx:^∞

n0

1

2ⁿφ2ⁿx. 3.35

It is clear thatfis continuous, bounded by 2 onCand f

xy

−fx−f

y≤6

|x|y 3.36

for allx, y∈Csee35. It follows from3.36that the following inequality:

f

xyz

−fx−f y

−fz≤12

|x|y|z|

3.37 holds for allx, y, z∈C.First we show that

fλx−λfx≤21|λ|²|x| 3.38

for allx, λ∈C.Iffsatisfies3.38for all|λ| ≥1,thenfsatisfies3.38for allλ∈C.To see this, let 0<|λ|<1the result is obvious whenλ0. Then|fλ⁻¹x−λ⁻¹fx| ≤21 |λ|⁻¹²|x|for allx∈C.Replacingxbyλx, we get that|fλx−λfx| ≤2|λ|²1 |λ|⁻¹²|x|21|λ|²|x|

for allx∈C.Hence we may assume that|λ| ≥1.Ifλx0 or|λx| ≥1,then

fλx−λfx≤21|λ|≤2|λ|1|λ||x| ≤21|λ|²|x|. 3.39

Now suppose that 0<|λx|<1.Then there exists an integerk≥0 such that

1

2^k1 ≤ |λx|< 1

2^k. 3.40

Therefore

2^k|x|, 2^k|λx| ∈−1,1. 3.41

Hence

2^m|x|, 2^m|λx| ∈−1,1 3.42

for allm0,1, . . . , k.From the definition offand3.40, we have

fλx−λfx

∞ nk1

1

2ⁿ φ2ⁿλx−λφ2ⁿx

≤1|λ| ^∞

nk1

1

2ⁿ 1|λ|

2^k ≤2|λ|1|λ||x| ≤21|λ|²|x|.

3.43

Thereforefsatisfies3.38. Now we prove that Dμf

x, y, z

−f

μxμyμz

≤

16|α|⁻¹1|α|²β⁻¹

1β²γ⁻¹

1γ² |x|y|z| 3.44 for allx, y, z∈Cand allμ∈T¹:{λ∈C:|λ|1},where

Dμf x, y, z

:αf

μxμy 2α

βf

μyμz 2β

γμf

zx 2γ

. 3.45

It follows from3.37and3.38that Dμf

x, y, z

−f

μxμyμz

≤ αf

μxμy 2α

−f

μxμy 2

βf

μyμz 2β

−f

μyμz 2

γμf zx

2γ

−μfzx 2

μfzx 2

−f

μzμx 2

f

μxμy 2

f

μyμz 2

f

μzμx 2

−f

μxμyμz

≤

6|α|⁻¹1|α|²xy

6β⁻¹

1β²yz

10γ⁻¹

1γ²

|xz|

≤

16|α|⁻¹1|α|²β⁻¹

1β²γ⁻¹

1γ²|x|y|z|

3.46

for allx, y, z∈Cand allμ∈T¹.Thusfsatisfies inequality3.18forp1.LetT :C → Cbe a linear functional such that

fx−Tx≤M|x| 3.47

for allx ∈ C,whereMis a positive constant. Then there exists a constantc ∈ Csuch that Tx cxfor all rational numbersx. So we have

fx≤M|c||x| 3.48

for all rational numbersx. Letm∈Nwithm > M|c|.Ifx0∈0,2^−m1∩Q, then 2ⁿx0∈0,1 for alln0,1, . . . , m−1.So

fx0≥^m−1

n0

1

2ⁿφ2ⁿx0 mx0>M|c|x0, 3.49

which contradicts3.48.

Acknowledgments

The third author was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology NRF-2009-0070788. The authors would like to thank the referees for a number of valuable suggestions regarding a previous version of this paper.

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